Abstract

Optical interferometric techniques offer non-contact, high accuracy and full field measurements, which are very attractive in various research and application fields. Single fringe pattern processing is often needed when measuring fast phenomenon. However, several difficulties are encountered in phase retrieval, among which the discontinuity problem of the fringe pattern is challenging and requires attention due to the increasing complexity of manufactured pieces. In this paper, we propose a complete flowchart for discontinuous single fringe pattern processing, which uses segmentation as universal pre-processing for all discontinuous fringe pattern problems. Within the flowchart, we also propose a systematic way to introduce boundary-awareness into demodulation methods by a masking function to improve demodulation accuracy and a quality-guided scanning strategy with a novel composite quality map to improve demodulation robustness. To the best of our knowledge, this is the first time a complete solution is proposed for single and discontinuous fringe pattern processing. Three typical demodulation methods, the frequency-guided regularized phase tracker with quadratic phase matching, the windowed Fourier ridges, and the spiral phase quadrature transform, are used to demonstrate how demodulation methods can be made boundary-aware. The proposed methods are verified by successful demodulation results from both simulated and experimental fringe patterns.

© 2017 Optical Society of America

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References

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  2. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001).
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    [PubMed]
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2016 (2)

B. Li, C. Tang, X. Zhu, Y. Su, and W. Xu, “Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity,” Opt. Lasers Eng. 78, 91–98 (2016).

H. Wang and Q. Kemao, “Local orientation coherence based segmentation and boundary-aware diffusion for discontinuous fringe patterns,” Opt. Express 24(14), 15609–15619 (2016).
[PubMed]

2015 (1)

2012 (3)

H. Wang and Q. Kemao, “Quality-guided orientation unwrapping for fringe direction estimation,” Appl. Opt. 51(4), 413–421 (2012).
[PubMed]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).

S. Li, X. Su, and W. Chen, “Hilbert assisted wavelet transform method of optical fringe pattern phase reconstruction for optical profilometry and interferometry,” Optik (Stuttg.) 123(1), 6–10 (2012).

2011 (2)

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).

J. Parkhurst, G. Price, P. Sharrock, and C. Moore, “Phase unwrapping algorithms for use in a true real-time optical body sensor system for use during radiotherapy,” Appl. Opt. 50(35), 6430–6439 (2011).
[PubMed]

2010 (1)

2009 (1)

2008 (3)

2007 (1)

2006 (2)

2001 (3)

1999 (1)

1996 (1)

Andresen, K.

Arnold, J. F.

Baird, J. P.

Bone, D. J.

Botella, A. G.

J. A. Quiroga, J. A. G. Pedrero, and A. G. Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1), 43–51 (2001).

Busca, G.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).

Chen, W.

S. Li, X. Su, and W. Chen, “Hilbert assisted wavelet transform method of optical fringe pattern phase reconstruction for optical profilometry and interferometry,” Optik (Stuttg.) 123(1), 6–10 (2012).

Cuevas, F. J.

Estrada, J. C.

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).

J. C. Estrada, M. Servín, J. A. Quiroga, and J. L. Marroquín, “Path independent demodulation method for single image interferograms with closed fringes within the function space C(2),” Opt. Express 14(21), 9687–9698 (2006).
[PubMed]

Galvan, C.

Gao, W.

Hock Soon, S.

Jueptner, W.

Kai, L.

Kemao, Q.

Larkin, K. G.

Li, B.

B. Li, C. Tang, X. Zhu, Y. Su, and W. Xu, “Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity,” Opt. Lasers Eng. 78, 91–98 (2016).

Li, K.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).

Li, S.

S. Li, X. Su, and W. Chen, “Hilbert assisted wavelet transform method of optical fringe pattern phase reconstruction for optical profilometry and interferometry,” Optik (Stuttg.) 123(1), 6–10 (2012).

Lin, F.

Marroquin, J. L.

Marroquín, J. L.

Moore, C.

Oldfield, M. A.

Osten, W.

Parkhurst, J.

Pedrero, J. A. G.

J. A. Quiroga, J. A. G. Pedrero, and A. G. Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1), 43–51 (2001).

Price, G.

Quiroga, J. A.

Rivera, M.

Seah, H. S.

Servin, M.

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001).

Servín, M.

Sharrock, P.

Su, X.

S. Li, X. Su, and W. Chen, “Hilbert assisted wavelet transform method of optical fringe pattern phase reconstruction for optical profilometry and interferometry,” Optik (Stuttg.) 123(1), 6–10 (2012).

Su, Y.

B. Li, C. Tang, X. Zhu, Y. Su, and W. Xu, “Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity,” Opt. Lasers Eng. 78, 91–98 (2016).

Tang, C.

B. Li, C. Tang, X. Zhu, Y. Su, and W. Xu, “Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity,” Opt. Lasers Eng. 78, 91–98 (2016).

Vargas, J.

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).

Wang, H.

Xu, W.

B. Li, C. Tang, X. Zhu, Y. Su, and W. Xu, “Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity,” Opt. Lasers Eng. 78, 91–98 (2016).

Yu, Q.

Zappa, E.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).

Zhao, M.

Zhou, X.

Zhu, X.

B. Li, C. Tang, X. Zhu, Y. Su, and W. Xu, “Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity,” Opt. Lasers Eng. 78, 91–98 (2016).

Appl. Opt. (8)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. A. Quiroga, J. A. G. Pedrero, and A. G. Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1), 43–51 (2001).

Opt. Express (2)

Opt. Lasers Eng. (4)

B. Li, C. Tang, X. Zhu, Y. Su, and W. Xu, “Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity,” Opt. Lasers Eng. 78, 91–98 (2016).

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).

Opt. Lett. (3)

Optik (Stuttg.) (1)

S. Li, X. Su, and W. Chen, “Hilbert assisted wavelet transform method of optical fringe pattern phase reconstruction for optical profilometry and interferometry,” Optik (Stuttg.) 123(1), 6–10 (2012).

Other (1)

Q. Kemao, Windowed Fringe Pattern Analysis (SPIE, 2013).

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Figures (14)

Fig. 1
Fig. 1 Flowchart of boundary-aware single fringe pattern processing (dotted squares indicate optional processing steps)
Fig. 2
Fig. 2 Simulated fringe patterns. (a) A simulated fringe pattern with linear boundary; (b) a simulated fringe pattern with circular boundary; (c) Fig. (a) with additive noise; (d) Fig. (b) with speckle noise.
Fig. 3
Fig. 3 Segmentation results of (a) Fig. 2(a); (b) Fig. 2(b); (c) Fig. 2(c); (d) Fig. 2(d).
Fig. 4
Fig. 4 Denoising results of (a) Fig. 2(c); (b) Fig. 2(d).
Fig. 5
Fig. 5 Background removal and amplitude normalization results of (a) Fig. 4(a); (b) Fig. 4(b).
Fig. 6
Fig. 6 Demodulation results of (a) Fig. 5(a) using QFGRPT; (b) Fig. 5(a) using BQFGRPT (wrapped); (c) Fig. 5(a) using BQFGRPT (unwrapped); (d) Fig. 5(b) using QFGRPT; (e) Fig. 5(b) using BQFGRPT (wrapped); (f) Fig. 5(b) using BQFGRPT (unwrapped).
Fig. 7
Fig. 7 Wrapped demodulation results of (a) Fig. 5(a) using WFR; (b) Fig. 5(a) using BWFR; (c) Fig. 5(b) using WFR; (d) Fig. 5(b) using BWFR.
Fig. 8
Fig. 8 Wrapped demodulation results of (a) Fig. 5(a) using SQT; (b) Fig. 5(a) using BSQT; (c) Fig. 5(b) using SQT; (d) Fig. 5(b) using BSQT.
Fig. 9
Fig. 9 Continuous demodulation results after boundary-aware unwrapping of (a) Fig. 7(b); (b) Fig. 7(d); (c) Fig. 8(b); (d) Fig. 8(d).
Fig. 10
Fig. 10 RMSEs of demodulation results of (a) Fig. 2(a); (b) Fig. 2(b); (c) Fig. 5(a); (d) Fig. 5(b).
Fig. 11
Fig. 11 Experimental fringe pattern. (a) An experimental fringe pattern from fringe projection profilometry; (b) the segmentation result; (c) the normalization result.
Fig. 12
Fig. 12 Demodulate results of Fig. 11(c). Wrapped phase of (a) BQFGRPT; (b) BWFR; (c) BSQT; unwrapped phase of (d) BQFGRPT; (e) BWFR; (f) BSQT;
Fig. 13
Fig. 13 Experimental fringe pattern. (a) An experimental fringe pattern from electronic speckle pattern shearing interferometry; (b) the segmentation result; (c) the normalization result.
Fig. 14
Fig. 14 Demodulate results of Fig. 13(c). Wrapped phase of (a) BQFGRPT; (b) BWFR; (c) BSQT; unwrapped phase of (d) BQFGRPT; (e) BWFR; (f) BSQT;

Equations (29)

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f( x,y )=a( x,y )+b( x,y )cos[ φ( x,y ) ]+n( x,y ),
T( x,y )=[ ( ε,η ) N x,y ρ( ε,η ) f xσ 2 ( ε,η ) ( ε,η ) N x,y ρ( ε,η ) f xσ ( ε,η ) f yσ ( ε,η ) ( ε,η ) N x,y ρ( ε,η ) f xσ ( ε,η ) f yσ ( ε,η ) ( ε,η ) N x,y ρ( ε,η ) f yσ 2 ( ε,η ) ],
D( x,y )=[ λ 2 ( x,y )>thr ],
f a ( x,y )=b( x,y )cos[ φ( x,y ) ];
f n ( x,y )=cos[ φ( x,y ) ].
S( x,y )=f( x,y )M( x,y ),
M( x,y )={ 1, ( x,y )s 0, ( x,y )s .
h( x,y )= ( ε,η ) N x,y r( ε,η ) ,
h B ( x,y )= ( ε,η ) N x,y M( ε,η )r( ε,η )
P( x,y )=[ Q( x,y )+ Q max ( ε,η ) N x,y M( ε,η ) ],
U( x,y )= ( ε,η ) N x,y ( { f n ( ε,η )cos[ φ e ( x,y;ε,η ) ] } 2 +δ [ φ 0 ( ε,η ) φ e ( x,y;ε,η ) ] 2 m( ε,η ) ) ,
φ e ( x,y;ε,η )= φ 0 ( x,y )+ ω x ( x,y )( εx )+ ω y ( x,y )( ηy ) + c xx ( x,y ) ( εx ) 2 /2+ c yy ( x,y ) ( ηy ) 2 /2+ c xy ( x,y )( εx )( ηy ),
ω( x,y )= ω x 2 ( x,y )+ ω y 2 ( x,y ) .
r( ε,η )= { f n ( ε,η )cos[ φ e ( x,y;ε,η ) ] } 2 +δ [ φ 0 ( ε,η ) φ e ( x,y;ε,η ) ] 2 m( ε,η ).
U B ( x,y )= ( ε,η ) N x,y M( ε,η )( { S n ( ε,η )cos[ φ e ( x,y,ε,η ) ] } 2 +δ [ φ 0 ( ε,η ) φ e ( x,y,ε,η ) ] 2 m( ε,η ) ) ,
Sf( u,v; ξ x , ξ y )= f a ( x,y ) g u,v; ξ x , ξ y ( x,y )dxdy ,
f a ( x,y )= 1 4 π 2 Sf( u,v; ξ x , ξ y ) g u,v; ξ x , ξ y ( x,y )d ξ x d ξ y dudv
g u,v; ξ x , ξ y ( x,y )=g( xu,yv )exp( j ξ x x+j ξ y y ) ,
g( x,y )= 1 π σ x σ y exp( x 2 2 σ x 2 y 2 2 σ y 2 ).
[ ω ax ( u,v ), ω ay ( u,v ) ]=arg max ξ x , ξ y | Sf( u,v; ξ x , ξ y ) |,
sign( u,v )={ 1 [ ω ax ( u,v ), ω ay ( u,v ) ][ ω x ( u p , v p ), ω y ( u p , v p ) ]0 1 otherwise ,
φ w ( u,v )=angle{ Sf[ u,v; ω x ( u,v ), ω y ( u,v ) ] }+ ω x ( u,v )u+ ω y ( u,v )v .
S f B ( u,v; ξ x , ξ y )= M( x,y )[ f a ( x,y ) g u,v; ξ x , ξ y ( x,y ) ]dxdy                         = S a ( x,y ) g u,v; ξ x , ξ y ( x,y )dxdy
V( f a )=iexp[ jϑ( x,y ) ] F 1 { exp[ jϕ( u,v ) ]F[ f a ( x,y ) ] }           =b( x,y )sin[ φ( x,y ) ]
φ w ( x,y )=arctan2[ V( f a ), f a ( x,y ) ]
θ( x,y )= 1 2 arctan2{ ( ε,η ) N x,y 2 f xσ ( ε,η ) f yσ ( ε,η ) , ( ε,η ) N x,y f yσ 2 ( ε,η ) f xσ 2 ( ε,η ) }.
ϑ( x,y )={ θ( x,y ) { cos[ θ( x,y ) ],sin[ θ( x,y ) ] }{ cos[ θ( x p , y p ) ],sin[ θ( x p , y p ) ] }0 M 2π [ θ( x,y )+π ] else ,
θ B ( x,y )= 1 2 arctan2{ ( ε,η ) N x,y 2M( ε,η ) S xσ ( ε,η ) S yσ ( ε,η ) , ( ε,η ) N x,y M( ε,η )[ S yσ 2 ( ε,η ) S xσ 2 ( ε,η ) ] } .
φ( x,y )= φ w ( x,y )+round{ [ φ( x p , y p ) φ w ( x,y ) ]/2 π }2π

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